OK, I think I just figured this out.

Rachael was on the right track, she just left out one step: I had to go to the Instrument dropdown and select "No Patch Change" for each instrument.

I saved the song as a template, but it turns out that I didn't even need to do that. Now if I open a new song, I get the same sounds and all the instruments are set to No Patch Change. Even when I closed the program and re open it, the settings are the same.

What a relief. I've been buried in this for a day and a half.

Thanks to both of you guys for the suggestions.